There is a proof that I'm trying to understand, but the textbook I have isn't making any sense and my teachers explanations are not very clear to non-existent. I'm trying to understand why: $$x^4 - y^4 = z^2$$ has no solutions. I've found a proof of this theorem here, but there is little to no explanation as to why each step was taken. I was hoping someone out there who understands this proof well can explain it to me, or provide an alternative way to prove it.
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I usually first prove that if $x^2+y^2=z^2$ for some integers, then the right-angled triangle with sides $x,y,z$ cannot have square area. You can find a proof of that here: Pythagorean triples and perfect squares. Then if there is an integer solution to your equation, you can deduce a right-angled triangle with integer sides and square area.
You can also find a direct proof here: Solving $x^4-y^4=z^2$